Explain the transformations of the graph y=2 3sqrtx+2-3 from its parent function y=x.
The parent function y=x is a linear function with a slope of 1 and a y-intercept of 0.
The transformed function y=2√(3x+2)-3 has several transformations applied to it:
1. Vertical stretching by a factor of 2: The coefficient of 2 in front of the square root term causes the graph to stretch vertically by a factor of 2.
2. Horizontal compression by a factor of 1/3: The coefficient of 3 inside the square root causes the graph to compress horizontally by a factor of 1/3.
3. Horizontal shift to the left by 2 units: The +2 added inside the square root term shifts the graph 2 units to the left.
4. Vertical shift down by 3 units: The -3 at the end of the function shifts the graph down by 3 units.
By combining these transformations, the graph of y=2√(3x+2)-3 is a vertically stretched, horizontally compressed, horizontally shifted left, and vertically shifted down version of the parent function y=x.